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A noncommutative version of Lie algebras: Leibniz algebras. (Une version non commutative des algèbres de Lie: les algèbres de Leibniz.) (French) Zbl 0806.55009
A Leibniz algebra over a commutative ring \(\mathbb{K}\) is a \(\mathbb{K}\)-module \(g\) together with a bilinear map \([-,-]: g\times g\to g\) satisfying for all \(x\), \(y\), \(z\) in \(G\), \([x, [y,z]] - [[x, y],z]+ [[x, z],y]=0\). The paper introduces this new algebraic structure and presents some interesting properties and connections with other theories. The main motivation is the existence of a homology theory \(HL_*\) that gives in particular new algebraic invariants for Lie algebras.
This paper is clearly written and gives a lot of new and interesting questions in homological algebra.

17A32 Leibniz algebras
17B55 Homological methods in Lie (super)algebras
18G60 Other (co)homology theories (MSC2010)
19D55 \(K\)-theory and homology; cyclic homology and cohomology